What is the quickest / best way of duplicating an object with children, so that the children are also duplicated?
You can do this by using the 'Select Grouped' menu, triggered with Shift+G.


Select Parent.
Shift+G> Children.
Shift select parent (by default it will be deselected when the children get selected).
Shift+D to duplicate.


If you are doing this multiple times you can open the toolbar(T) and check 'Extend' after doing 'Select Grouped', so the parent remains selected when the children are selected. Any future selection of the children will respect this 'Extend' setting.